The orientation of a sphere is an element of and can be represented by three Euler angles. However, uniformly sampling three Euler angles does not result in a uniform sampling of . To generate a uniformly distributed random rotation in , first perform a random rotation about the axis, then rotate the axis to a random position on the sphere.
We can use this same method to generate uniformly distributed perturbations in by scaling the axis rotation and deflection by a number less than 1.
This Demonstration provides a fast way to generate observations from a uniform distribution on . It also provides a way to generate uniformly distributed perturbations about the axis, which may be useful for simulations and sampling-based planners.
 J. Arvo, "Fast Random Rotation Matrices," Graphics Gems III, 1991.
Arvo's code results in a rotation of about the world axis, leading to incorrect results when the code is used to sample a perturbation. This Demonstration removes that error by premultiplying Arvo's result by a rotation of about the world axis. This flips the signs of st and ct, which flips the sign of the first two columns of the rotation matrix. This Demonstration also centers the perturbations about zero instead of biasing the result.